Polar form

If \(z=a+bi\neq0\), the point \((a,b)\) is determined by its distance \(r=|z|\) to the origin and by the angle \(\theta\) that the segment makes with the positive real axis. From trigonometry, \(a=r\cos\theta\) and \(b=r\sin\theta\), which gives the formula. To recover \(\theta\) use \(\tan\theta=b/a\) while checking the quadrant, not just the arctangent.

The argument is not unique: if \(\theta\) works, so does \(\theta+2k\pi\) for any integer \(k\). To fix one we take the principal argument \(\operatorname{Arg}z\in(-\pi,\pi]\). For example, \(1+i=\sqrt2\left(\cos\frac\pi4+i\sin\frac\pi4\right)\) and \(-1=\cos\pi+i\sin\pi\). Zero is excluded because its angle is undetermined.

The payoff is in the product: if \(z_1=r_1(\cos\theta_1+i\sin\theta_1)\) and \(z_2=r_2(\cos\theta_2+i\sin\theta_2)\), the angle-sum identities give \(z_1z_2=r_1r_2\big(\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)\big)\). So multiplying scales the moduli and adds the angles; dividing subtracts the angles. Multiplying by \(i\) rotates by \(\pi/2\).

Later \(\cos\theta+i\sin\theta\) is abbreviated \(e^{i\theta}\), and the polar form becomes \(z=re^{i\theta}\). Watch out for two mistakes: giving a negative modulus and adding the moduli instead of multiplying them.